Interpretable Machine Learning for Spatial Science: A Lie-Algebraic Kernel for Rotationally Anisotropic Gaussian Processes
Authors: Kane Warrior, Dalia Chakrabarty
Summary
The authors introduce a Gaussian process kernel for 3D spatial data that explicitly parameterizes rotated anisotropy: three principal length-scales plus an SO(3) rotation encoded as an axis–angle vector. Standard ARD kernels only capture axis-aligned stretching; generic symmetric-positive-definite parameterizations can represent arbitrary rotations but don't expose the principal directions. Here the rotation is mapped via the Lie-algebra exponential, giving unconstrained Euclidean coordinates for inference while guaranteeing a valid covariance metric. Bayesian MCMC on synthetic and real nano-material density data shows the method recovers the true generating metric, improves prediction over ARD when the field is rotated, and matches ARD when the field is axis-aligned.
Main takeaways:
- Exposes three principal length-scales and an explicit SO(3) rotation as interpretable parameters, unlike black-box full-SPD matrices
- Uses axis–angle Lie-algebra exponential to keep the rotation unconstrained during optimization while always producing a valid rotation matrix
- Posterior inference via MCMC; the paper characterizes symmetries and weakly identified regimes (e.g., when two length-scales are similar)
- On synthetic rotated-anisotropic data, recovers the ground-truth metric and outperforms axis-aligned ARD; matches ARD when data is axis-aligned
- Applied to a nano-brick material-density dataset, the inferred metric reveals rotated principal directions
Relevance
Not related to my persona or midtraining work—this is a spatial-statistics method for physical/material science data. Included because it's a useful interpretable-GP reference if I ever need structured anisotropic priors.
Abstract
arXiv:2605.11179v1 Announce Type: new Abstract: Many three-dimensional spatial fields are anisotropic, with directions of rapid and slow variation that need not align with the coordinate axes. Standard Gaussian process kernels with Automatic Relevance Determination (ARD) capture only axis-aligned anisotropy, while generic full symmetric positive definite (SPD) metrics can represent rotated anisotropy but do not parameterise principal length-scales and directions directly. We introduce an interpretable rotationally anisotropic GP kernel that parameterises a three-dimensional SPD covariance metric using three principal length-scales and an explicit SO(3) rotation. The rotation is represented by an axis-angle vector and mapped to SO(3) via the Lie-algebra exponential map, giving unconstrained Euclidean coordinates for inference while always inducing a valid SPD metric. The construction spans the same family of three-dimensional SPD covariance metrics as a generic full-SPD parameterisation, but exposes the geometry differently: length-scales and orientation are explicit, interpretable, and directly available for prior specification and posterior summaries. We perform Bayesian inference on these quantities using Markov Chain Monte Carlo (MCMC), and characterise the resulting symmetries and weakly identified regimes. On synthetic data with rotated anisotropy, the posterior recovers the generating metric and improves prediction relative to an axis-aligned ARD baseline, while matching the predictive performance of a generic full SPD baseline. When the ground truth is axis-aligned, posterior mass concentrates near the identity rotation and predictive performance matches ARD. On a material-density dataset from a laboratory-fabricated nano-brick, the inferred metric reveals rotated anisotropy that is not captured by axis-aligned kernels.